Max Planck Institute for Gravitational Physics (Albert Einstein Institute)
D-14476 Potsdam, Germany
VisualLie is a web app to visualize the root system of the Feingold-Frenkel algebra.
The Feingold-Frenkel algebra $\mathcal{F}$ (or $AE_3$ or $A_1^{++}$) is a rank 3 hyperbolic Kac-Moody algebra associated with the indefinite Cartan matrix \begin{align}\label{eq:CM} A = (a_{ij}) = \begin{pmatrix}2 & -1 & 0 \\ -1 & 2 & -2 \\ 0 & -2 & 2 \end{pmatrix} \, . \end{align} Its study was pioneered in [1]. Below the figures, we give a more comprehensive introduction to the Feingold-Frenkel algebra $\mathcal{F}$ and, in particular, highlight some of the properties of its root system.
The figure shows the root system of $\mathcal{F}$ for one level at a time. The notion of level is explained below the figure. The figure is fully interactive. Hover over a root to see its root vector, norm, and multiplicity. Click on a root to see its Weyl orbit. Use the bokeh plot tools on the right to pan, zoom, save, or reset the plot. Use the selectors, checkboxes, and buttons on the left to change the level of the visualized roots and turn on additional plot features.
Instead of just plotting one level at the time we can also plot the entire root system at once. The resulting picture for the levels 1 to 3 and up to depth 15 is shown below. The roots are colored according to their level. Also this plot is fully interactive. Use your mouse to move around the root system and to zoom in and out. With the button on the right hand side a hyperboloid can be added to the picture. Notice that the entire root system fits into the hyperboloid. The reason for this is the same why all the Weyl orbits above are parabolas. We explain it in a bit more detail in section 6 below.
In the following, we mostly use the notation and conventions of [2]. The matrix $\small{\eqref{eq:CM}}$ is a so-called generalized Cartan matrix. As such, it satisfies the following conditions:
We call a Cartan matrix indecomposable if there is no decomposition of the sets of indices $I$ into a disjoint union of non-empty subsets $I_1$ and $I_2$ such that $a_{ij} = 0$ for all $i$ in $I_1$ and $j$ in $I_2$. A Cartan matrix $C$ is called symmetrizable if there exists a diagonal matrix $D$ (with positive integer entries) and a symmetric matrix $S$ such that $C = DS$. Assuming that C is symmetrizable and indecomposable, there are three types of generalized Cartan matrices:
For Cartan matrices of indefinite type, there is a special subclass. A generalized Cartan matrix is said to be of hyperbolic type if it is of indefinite type, but for any proper subset of $I$, the corresponding submatrix is of finite or affine type. In particular, $\small{\eqref{eq:CM}}$ is of hyperbolic type.
A realization of $\small{\eqref{eq:CM}}$ is a triple $(\mathfrak{h}, \Pi, \Pi^\vee)$, where $\mathfrak{h}$ is a complex vector space and $\Pi = \{ \boldsymbol{\alpha}_{-1}, \boldsymbol{\alpha}_0, \boldsymbol{\alpha}_1 \} \subset \mathfrak{h}^\ast$ and $\Pi^\vee = \{ \boldsymbol{\alpha}_{-1}^\vee, \boldsymbol{\alpha}_0^\vee, \boldsymbol{\alpha}_1^\vee \} \subset \mathfrak{h}$ are indexed subsets in $\mathfrak{h}^\ast$ and $\mathfrak{h}$ respectively, satisfying the following conditions:
Here, $\langle \cdot, \cdot \rangle$ is the usual pairing between a vector space and its dual. The elements of $\Pi$ are called simple roots, and the elements of $\Pi^\vee$ are called simple co-roots. They form the root (respectively co-root) basis. In the following, we choose \begin{align}\label{eq:SimpleRoots} \boldsymbol{\alpha}_{-1} = (1,0,0) \, , \quad \boldsymbol{\alpha}_0 = (0,1,0)\, , \quad \boldsymbol{\alpha}_1=(0,0,1) \end{align} and find \begin{align} \boldsymbol{\alpha}_{-1}^\vee = (2,-1,0) \, , \quad \boldsymbol{\alpha}_0^\vee = (-1,2,-2)\, , \quad \boldsymbol{\alpha}_1^\vee =(0,-2,2) \, . \end{align} $Q = \mathbb{Z}_{-1} \boldsymbol{\alpha}_{-1} + \mathbb{Z}_0 \boldsymbol{\alpha}_0 + \mathbb{Z}_1 \boldsymbol{\alpha}_1$ is called the root lattice of $\mathcal{F}$.
Let $\mathfrak{g} = \mathfrak{g}(A)$ be a hyperbolic Kac-moody algebra associated with the Cartan matrix $\small{\eqref{eq:CM}}$. Then, its generators $\{e_i, f_i, h_i\}$ obey the commutation relations \begin{align} \begin{aligned} &[h_i, h_j] = 0 \, , \quad \quad \quad [e_i,f_j] = \delta_{ij} h_i\\ &[h_i, e_j] = a_{ij} \, e_j \, , \quad \ \, [h_i,f_j] = - a_{ij} \, f_j \, , \\[1.5ex] &\mathrm{ad}(e_i)^{1-a_{ij}}(e_j) = \mathrm{ad}(f_i)^{1-a_{ij}}(f_j) ,= 0\, . \end{aligned} \end{align} In particular $h_i \in \mathfrak{h}$ for $i=-1,0,1$. The algebra $\mathfrak{g}(A)$ can then be decomposed into root spaces with respect to $\mathfrak{h}$ \begin{align} \mathfrak{g}(A) = \bigoplus_{\boldsymbol{\alpha} \in Q} \mathfrak{g}_{\boldsymbol{\alpha}} \, . \end{align} Here $\mathfrak{g}_{\boldsymbol{\alpha}} = \{ x \in \mathfrak{g}(A) \ | \ [h,x] = \alpha(h) x \ \text{for all } h \in \mathfrak{h}\}$ is the root space attached to $\boldsymbol{\alpha}$, with $\mathfrak{g}_0 = \mathfrak{h}$. The number $\alpha = \mathrm{dim}(\mathfrak{g}_{\boldsymbol{\alpha}})$ is called the multiplicity of $\boldsymbol{\alpha}$. An element $\boldsymbol{\alpha} \in Q$ is called a root if $\boldsymbol{\alpha} \not=0$ and $\mathrm{mult}(\boldsymbol{\alpha}) \not = 0$. We denote the set of all roots by $\Delta$. Every root is either positive (i.e. $\boldsymbol{\alpha} > 0$) or negative (i.e. $\boldsymbol{\alpha} < 0$). Hence, the set of all roots $\Delta$ can be written as the disjoint union of the set of positive roots $\Delta_+$ and the set of negative roots $\Delta_-$. The simple roots are in $\Delta_+$, and they all have multiplicity one.
A $\mathfrak{g}(A)$-module $V$ is called $\mathfrak{h}$-diagonalizable if $V = \bigoplus_{\lambda \in \mathfrak{h}^\ast} V_\lambda$, where $V_\lambda = \{ v \in V \ | \ h(v) = \langle \lambda, h \rangle v \ \text{for} \ h \in \mathfrak{h} \}$. Then $V_\lambda$ is called weight space, $\lambda \in \mathfrak{h}^\ast$ is called a weight if $V_\lambda \not= 0$, and $\mathrm{dim}(V_\lambda)$ is called the multiplicity of $\lambda$ and is denoted by $\mathrm{mult}_V(\lambda)$. By $P(V) = \{\lambda \in \mathfrak{h}^\ast \ | \ V_\lambda \not=0\}$ we denote the set of weights of $V$.
A module $V$ is called a highest weight module with highest weight $\boldsymbol{\Lambda} \in \mathfrak{h}^\ast$ if there exist a non-zero vector $v_{\boldsymbol{\Lambda}} \in V$ such that $e_i(v_{\boldsymbol{\Lambda}}) = 0$, $h_i(v_{\boldsymbol{\Lambda}}) = \boldsymbol{\Lambda}(h) v_{\boldsymbol{\Lambda}}$ for all $i = -1, 0, 1$ and if $U(\mathfrak{g}(A))(v_{\boldsymbol{\Lambda}}) = V$, where $U$ is the universal enveloping algebra. The vector $v_{\boldsymbol{\Lambda}}$ is called a highest weight vector.
An $\mathfrak{h}$-diagonizable module $V$ over a Kac-Moody algebra $\mathfrak{g}(A)$ is called integrable if all $e_i$ and $f_i$ ($i=-1,0,1$) are locally nilpotent on $V$. In the following, we denote integrable highest-weight modules by $L(\boldsymbol{\Lambda})$. It can be shown that the $\mathfrak{g}(A)$-module $L(\boldsymbol{\Lambda})$ is integrable if and only if $\boldsymbol{\Lambda} \in P_+$ with $P_+ = \{ \lambda \in \mathfrak{h}^\ast \ | \ \langle \lambda , \boldsymbol{\alpha}_i^\vee \rangle \in \mathbb{Z}_{\ge 0} \ \text{for all} \ i = -1, 0, 1\}$.
Let $\lambda \in \mathfrak{h}^\ast$. For each $i = -1, 0, 1$ we define the fundamental Weyl reflection $r_i$ of the space $\mathfrak{h}^\ast$ by \begin{align} r_i(\lambda) = \lambda - \left< \lambda, \boldsymbol{\alpha}_i^\vee \right> \boldsymbol{\alpha}_i \, . \end{align} The subgroup $W \subset \mathrm{GL}(\mathfrak{h}^\ast)$ generated by all fundamental reflections is called the Weyl group of $\mathfrak{g}(A)$.
A root $\boldsymbol{\alpha} \in \Delta$ is called real if there exists $w \in W$ such that $w(\boldsymbol{\alpha})$ is a simple root. For the Feingold-Frenkel algebra $\mathcal{F}$ all real roots have norm 2, i.e. $ (\boldsymbol{\alpha} , \boldsymbol{\alpha} ) = 2$. Here $(\cdot, \cdot)$ is the symmetric non-degenerate bilinear form defined by $(\boldsymbol{\alpha}_i, \boldsymbol{\alpha}_j) = a_{ij}$.
The following statements about real roots $\boldsymbol{\alpha}$ are important for constructing the root system of $\mathcal{F}$:
A root that is not real is called imaginary. Imaginary roots have norm $\le 0$. From the statements above, it follows in particular that if $\boldsymbol{\beta} \in \Delta$ also $w(\boldsymbol{\beta}) \in \Delta$ for all $w \in W$. Moreover, it can be shown that $\boldsymbol{\beta} \in Q$ is a root if and only if $|\boldsymbol{\beta}| \le 2$.
Starting from the set of simple roots $\Pi$, we can now construct the entire root system of $\mathcal{F}$. Roots that are connected through Weyl transformations have the same multiplicity. For all other roots, we must compute the multiplicity using the Weyl character formula. This is described in more detail in section 7 below.
A general root $\boldsymbol{\alpha}$ has root vector $\boldsymbol{\alpha} = a_{-1} \boldsymbol{\alpha}_{-1} + a_0 \boldsymbol{\alpha}_0 + a_1 \boldsymbol{\alpha}_1$ with $a_i \in \mathbb{Z}$ and either all $a_i \ge 0$ or all $a_i \le 0$. We call $-a_{-1}$ the level of a root; $-a_0$ is the depth, and $a_0 - a_1$ is the spin label (or weight). So, positive level and depth are associated with negative roots. In the first figure above, we sliced the root system into planes of equal level and only considered the negative roots. The two axes of the plot are the depth and spin label of the roots. The picture with negative level is obtained from a reflection of the plot across the x-axis. When hovering the mouse over a root, the root vector is shown without the minus signs to improve the readability. The second figure we consider the entire root system with the roots of different levels stacked on top of each other in the $z$ direction.
To classify the integrable highest weight representations, we introduce the fundamental weights $\boldsymbol{\Lambda}_i$ satisfying $\langle \boldsymbol{\Lambda}_i , \boldsymbol{\alpha}_j^\vee \rangle = \delta_{ij}$ such that \begin{align} \boldsymbol{\Lambda}_{-1} = - \boldsymbol{\delta} \, , \quad \boldsymbol{\Lambda}_0 = - \boldsymbol{\alpha}_{-1} - 2 \boldsymbol{\delta} \, , \quad \boldsymbol{\Lambda}_1 = - \boldsymbol{\alpha}_{-1} - 2 \boldsymbol{\delta} + \frac{1}{2} \boldsymbol{\alpha}_1 \, , \end{align} where $\boldsymbol{\delta} = \boldsymbol{\alpha}_0 + \boldsymbol{\alpha}_1$ is the so-called affine null root. It follows that the level $\ell$ of a root $\boldsymbol{\alpha}$ is $\ell = (\boldsymbol{\alpha}, \boldsymbol{\delta})$. The fundamental weights form a basis of the weight lattice $P = \mathbb{Z}_{-1} \boldsymbol{\Lambda}_{-1} + \mathbb{Z}_0 \boldsymbol{\Lambda}_0 + \mathbb{Z}_1 \boldsymbol{\Lambda}_1$. The highest weight modules at level $\ell$ are then given by \begin{align} L(\boldsymbol{\Lambda}) = L(p \boldsymbol{\Lambda}_0 + 2 q \boldsymbol{\Lambda}_1 + r \boldsymbol{\Lambda}_{-1}) \quad \text{with} \quad p + 2q = \ell \quad \text{and} \quad p,q,r \in \mathbb{Z}_{\ge 0} \, . \end{align} Here we have $2q$ so that $\boldsymbol{\Lambda} \in Q$. We show that $L(\boldsymbol{\Lambda})$ is integrable and that the highest weight $\boldsymbol{\Lambda} \in \Delta$. Clearly $\boldsymbol{\Lambda} \in \mathfrak{h}^\ast$. Moreover, $\langle \boldsymbol{\Lambda} , \boldsymbol{\alpha}_i^\vee \rangle \in \mathbb{Z}_{\ge 0}$ because $p,q,r \in \mathbb{Z}_{\ge 0}$. Finally it is not hard to check that $|\boldsymbol{\Lambda}| \le 0$, hence $\boldsymbol{\Lambda} \in \Delta$. Since the fundamental weights form a basis of the weight lattice, these are, in fact, all the highest weight modules at level $\ell$.
In [1], it was shown that there exists a decomposition of $\mathcal{F}$ with respect to a rank 2 affine subalgebra. Consider the generalized Cartan matrix \begin{align} \begin{pmatrix} 2 & -2 \\ -2 & 2 \end{pmatrix} \end{align} and the associated affine subalgebra $A_1^{(1)}$ of $\mathcal{F}$ generated by $\{e_i, f_i, h_i \ | \ i = 1, 2 \}$. An explicit description of $A_1^{(1)}$ is given by \begin{align} A_1^{(1)} = \mathrm{span}_\mathbb{R} \left\{ E_m, F_m, H_m \, (m \in \mathbb{Z}) , K , \mathfrak{d} \right\} \end{align} with the standard commutation relations \begin{align} \begin{aligned} &[H_m, E_m] = 2 E_{m+n}, &&[H_m, F_m] = -2F_{m+n} \, , \\ &[E_m, F_n] = H_{m+n} + m K \delta_{m+n,0} \, , &&[H_m, H_n] = 2m K \delta_{m+n,0} \, , \\ &[E_m, E_n] = [F_m, F_n] = 0 \, . \end{aligned} \end{align} Furthermore, \begin{align} K = h_0 + h_1 \, , \quad \mathfrak{d} = h_{-1} + h_0 + h_1 \, , \end{align} where $K$ commutes with all affine generators and $\mathfrak{d}$ is the depth counting operator with \begin{align} [\mathfrak{d}, T_m] = - m T_m \quad \text{for all} \quad T_m \in \{E_m, F_m, H_m\} \, . \end{align} Together with $h_1$, the operators $K$ and $\mathfrak{d}$ span the Cartan subalgebra $A_1^{(1)}$ of $\mathcal{F}$. After these preparations the algebra $\mathcal{F}$ can be decomposed into eigenspaces of $K$ \begin{align} \mathcal{F} = \bigoplus_{\ell \in \mathbb{Z}} \mathcal{F}^{(\ell)} \end{align} In particular $\mathcal{F}^{(0)} = A_1^{(1)}$ and $\mathcal{F}^{(1)} = L(\boldsymbol{\Lambda}_0)$. The highest weight vector in $L(\boldsymbol{\Lambda}_0)$ is $\boldsymbol{\Lambda}_0 + 2 \boldsymbol{\delta} = - \boldsymbol{\alpha}_{-1}$. This is the level decomposition of $\mathcal{F}$, and it is exactly what is shown in the figures above. Each subspace $\mathcal{F}^{(\ell)}$ decomposes into (generally infinitely many) irreducible representations of $A_1^{(1)}$. The subspace $\mathcal{F}^{(-\ell)}$ is conjugate to $\mathcal{F}^{(\ell)}$ and thus needs not be studied separately. The level-$\ell$ subspace $\mathcal{F}^{(\ell)}$ is spanned by the multi-commutators $[f_{i_1}, \ldots , [f_{i_{n-1}}, f_{i_n}] \ldots]$ with $\ell$ generators $f_{-1}$. Subsequently, it can be shown that [1] \begin{align} \mathcal{F}^{(\ell)} = \left[ \mathcal{F}^{(1)} , \mathcal{F}^{(\ell-1)} \right] \, . \end{align}
In this section we discuss the Weyl group $\hat{W}$ of the affine subalgebra $A_1^{(1)}$ of $\mathcal{F}$. $\hat{W}$ is generated by the fundamental reflections $r_0$ and $r_1$ acting on $\lambda \in \mathfrak{h}^\ast$ \begin{align} r_i(\lambda) = \lambda - \left< \lambda, \boldsymbol{\alpha}_i^\vee \right> \boldsymbol{\alpha} _i \, . \end{align} Two features are built into the 2d figure to understand the action of the Weyl group $\hat{W}$. Clicking on a root highlights all roots (up to the maximum depth of the plot) in the Weyl orbit of that root. Moreover, a line is drawn through the roots. Feingold-Frenkel have shown that this line is always a parabola. For a root $\boldsymbol{\alpha} = a_{-1} \boldsymbol{\alpha}_{-1} + a_0 \boldsymbol{\alpha}_0 + a_1 \boldsymbol{\alpha}_1$ the parabola is described through the equation \begin{align} f(x) = - \frac{1}{a_{-1}} \left( x^2 + a_{-1} a_0 - (a_0 - a_1)^2 \right) \, . \end{align} In the case of the full root system the analog of this equation tells us that all roots fit into a hyperbola.
When clicking the Weyl reflection checkbox, two lines are drawn into the root lattice. They visualize the fundamental Weyl reflections. The reflection $r_1$ amounts to a horizontal reflection across the line labeled $r_1$. Similarly, the reflection $r_0$ amounts to a diagonal reflection across the line labeled $r_0$. Notice that the $r_0$ line moves to the right as we increase the level. The direction of these reflections is also indicated by the arrows in the top left corner of the 2d figure.
There is one decomposition of the Weyl group for which the calculation of Weyl orbits becomes particularly simple. Let $t(\lambda) = r_1 (r_0(\lambda))$. Then $T = \{t^n \ | \ n \in \mathbb{Z} \}$ is called the group of translations. It can then be shown that \begin{align} \hat{W} = \{ r_1 \} \ltimes T \, . \end{align} In the plot, one can switch between the full Weyl group orbit and the translation group orbit with the buttons on the left-hand side. Interestingly, these two orbits start to differ only from level 3 onward.
In this section, we explain how the root multiplicities are calculated.
A useful way to decode the entire information of an integrable highest-weight module into a compact expression is the formal character formula \begin{align} \mathrm{ch}(L(\boldsymbol{\Lambda})) = \sum_{\lambda \in P(\boldsymbol{\Lambda})} \mathrm{mult}(\lambda) e^\lambda \, . \end{align} This formula is in one-to-one correspondence with the figures above. For example the level 1 of $\mathcal{F}$ is given by $\mathcal{F}^{(1)} = L(\boldsymbol{\Lambda}_0)$ and its character is \begin{align} \begin{aligned} \mathrm{ch}(\mathcal{F}_1) = \mathrm{ch} (L(\boldsymbol{\Lambda}_1)) &= 1 \cdot e^{-\boldsymbol{\alpha}_{-1}} + 1 \cdot e^{-\boldsymbol{\alpha}_{-1} - \boldsymbol{\alpha}_0} + 1 \cdot e^{-\boldsymbol{\alpha}_{-1} - \boldsymbol{\alpha}_0 - \boldsymbol{a}_1} + 1 \cdot e^{-\boldsymbol{\alpha}_{-1} - \boldsymbol{\alpha}_0 - 2 \boldsymbol{\alpha}_1} \\ &\quad + 1 \cdot e^{-\boldsymbol{\alpha}_{-1} - 2 \boldsymbol{\alpha}_0 - \boldsymbol{\alpha}_1} + 2 \cdot e^{-\boldsymbol{\alpha}_{-1} - 2 \boldsymbol{\alpha}_0 - 2 \boldsymbol{\alpha}_1} + \ldots \end{aligned} \end{align} To compute the multiplicities, we need to introduce some additional notation. Let $w = r_{1_1} \ldots r_{i_s} \in W$ be an element of the Weyl group and $\epsilon(w) = (-1)^{\ell(w)}$. Here $\ell(w) = s$ is the length of $w$, i.e., the minimal number of repeated Weyl reflections $r_i$ one has to do to get $w$. Moreover, let $\boldsymbol{\rho} = (1,1,1)$ be the Weyl vector of $\mathfrak{g}(A)$. Then the Weyl-Kac character formula is [3] \begin{align}\label{eq:CharacterFormula} \mathrm{ch}(L(\boldsymbol{\Lambda})) = \frac{\sum\limits_{w \in W} \epsilon(w) e^{-w(\boldsymbol{\Lambda}+\boldsymbol{\rho}) + \boldsymbol{\rho}}}{\prod\limits_{\boldsymbol{\alpha} \in \Delta_+} (1-e^{\boldsymbol{\alpha}})^{\mathrm{mult}({\boldsymbol{\alpha})}}} \, . \end{align} A particularly simple case arises when we specialize to the 1-dimensional trivial representation $L(0) = \mathbb{C}$. The resulting formula is known as the denominator formula \begin{align}\label{eq:DenominatorFormula} \sum_{w \in W} \epsilon(w) e^{-w(\boldsymbol{\rho})+\boldsymbol{\rho}} = \prod_{\boldsymbol{\alpha} \in \Delta_+} (1-e^{\boldsymbol{\alpha}})^{\mathrm{mult}(\boldsymbol{\alpha})} \, . \end{align} By expanding both sides and comparing coefficients, we can, in principle, obtain the multiplicities $\mathrm{mult}(\boldsymbol{\alpha})$ of all the root $\boldsymbol{\alpha} \in \Delta_+$. However, there is a more efficient and elegant way due to Peterson that we will outline below.
We call the denominator of $\small{\eqref{eq:CharacterFormula}}$ \begin{align} R= \prod_{\boldsymbol{\alpha} \in \Delta_+} (1-e^{\boldsymbol{\alpha}})^{\mathrm{mult}(\boldsymbol{\alpha})} \end{align} and introduce the so-called co-multiplicity \begin{align} c_{\boldsymbol{\alpha}} = \sum_{k \ge 1} \frac{1}{k} \, \mathrm{mult} \left( \frac{\boldsymbol{\alpha}}{k} \right) \, . \end{align} Here the sum is only over those $k \in \mathbb{Z}_{\ge0}$, where $\boldsymbol{\alpha}/k \in Q$. Then, it is not hard to show that \begin{align} F = - \log R= \sum_{\boldsymbol{\beta}>0} c_{\boldsymbol{\beta}} e^{\boldsymbol{\beta}} \, , \end{align} Here the sum is no longer over the roots $\alpha \in \Delta_+$ but over all root multiples, i.e. all $\boldsymbol{\beta} \in Q$ such that $k \boldsymbol{\alpha} = \boldsymbol{\beta}$ for some $\boldsymbol{\alpha} \in \Delta_+$ and $k \in \mathbb{Z}_{\ge0}$. Root multiples can have co-multiplicity $> 0$, but their multiplicity is always zero.
Then we choose an orthonormal basis $u_i$ of the Cartan subalgebra and take the Laplacian $\sum_i \partial_i^2$ (where $\partial_i e^{\boldsymbol{\alpha}} = \boldsymbol{\alpha}(u_i)e^{\boldsymbol{\alpha}}$) of the denominator formula $\small{\eqref{eq:DenominatorFormula}}$ \begin{align} \sum_i \partial_i^2 R = 2 \sum_i \boldsymbol{\rho}(u_i) \partial_i R \, . \end{align} This implies \begin{align} \sum_i \left( (\partial_i F)^2 - \partial_i^2 F \right) = - 2 \sum_{\boldsymbol{\alpha}>0} c_{\boldsymbol{\alpha}} (\boldsymbol{\alpha} , \boldsymbol{\rho}) e^{\boldsymbol{\alpha}} \, . \end{align} Expanding the left-hand side yields \begin{align} \sum_i \left( (\partial_i F)^2 - \partial_i^2 F \right) = \sum_i \left( \sum_{\boldsymbol{\beta} > 0} c_{\boldsymbol{\beta}} \, \boldsymbol{\beta}(u_i) e^{\boldsymbol{\beta}} \right) \left( \sum_{\boldsymbol{\gamma} > 0} c_{\boldsymbol{\gamma}} \boldsymbol{\gamma}(u_i) e^{\boldsymbol{\gamma}} \right) - \sum_{\boldsymbol{\alpha} > 0} c_{\boldsymbol{\alpha}} (\boldsymbol{\alpha}, \boldsymbol{\alpha}) e^{\boldsymbol{\alpha}} \end{align} and by comparing coefficients, we arrive at the \textbf{Peterson recursion formula} \begin{align}\label{eq:PetersonFormula} (\boldsymbol{\alpha},\boldsymbol{\alpha} - 2 \boldsymbol{\rho}) c_\alpha = \sum_{\substack{\boldsymbol{\beta}, \boldsymbol{\gamma} > 0 \\ \boldsymbol{\beta} +\boldsymbol{\gamma} = \boldsymbol{\alpha}}} c_{\boldsymbol{\beta}} c_{\boldsymbol{\gamma}} \, (\boldsymbol{\beta} , \boldsymbol{\gamma}) \, . \end{align} This formula tells us that we obtain the multiplicity of a root $\boldsymbol{\alpha}$ from the sum of the co-multiplicities of all root multiplets $\boldsymbol{\beta}$ and $\boldsymbol{\gamma}$, where $\boldsymbol{\beta} + \boldsymbol{\gamma} = \boldsymbol{\alpha}$. Thus, the steps to find new roots in section 3 together with $\small{\eqref{eq:PetersonFormula}}$ allow us to iteratively construct the root system of the Feingold-Frenkel algebra $\mathcal{F}$ starting from the set of simple roots $\small{\eqref{eq:SimpleRoots}}$.
At lower levels, it is possible to significantly improve the recursive construction of the root system.
Having a closer look at the level 1 of $\mathcal{F}$, we notice that the multiplicities in each string of roots extending into the $-y$ direction are the same. This is because the Weyl orbit of the highest weight $-\boldsymbol{\alpha}_{-1}$ contains the ends of all these root strings. It has been shown that the multiplicities in these root strings are nothing but the integer partitions. Hence, we can obtain them all from the generating function \begin{align}\label{eq:phi} \varphi(q) = \prod_{i \ge 1} \frac{1}{(1-q^n)} = 1 + q + q^2 + 2 q^3 + 3 q^3 + 5 q^4 + \ldots \end{align} We call $\varphi(q)$ a string function. Subsequently we can write the character of $\mathcal{F}_1 = L(\Lambda_1)$ as \begin{align}\label{eq:chF1} \mathrm{ch}(\mathcal{F}_1) = \mathrm{ch}(L(\boldsymbol{\Lambda_1})) = \varphi(q) \, \Theta_{\boldsymbol{\Lambda_1}} \, , \quad \text{with} \quad q = e^{-\boldsymbol{\delta}} \end{align} and where for any weight $\lambda \in \mathfrak{h}^\ast$ of level $\ell$ \begin{align} \Theta_\lambda = e^{- \frac{|\lambda|^2}{2\ell} \boldsymbol{\delta}} \sum_{t \in T} e^{t(\lambda)} \, . \end{align} At higher levels, the equation $\small{\eqref{eq:chF1}}$ becomes much more complicated. Already at level 2, we need two string functions to describe the character of $V(2\Lambda_1)$, and neither of them is as simple as $\small{\eqref{eq:phi}}$. To describe the character of $\mathcal{F}_2$, a total of four string functions are needed.
In general, one can write for any highest-weight representation \begin{align} \mathrm{ch}\left( L ( i \boldsymbol{\Lambda}_0 + j \boldsymbol{\Lambda}_1) \right) = \sum_{m,n} C_{ij}^{mn} \ \Theta_{m \boldsymbol{\Lambda}_0 + n \boldsymbol{\Lambda}_1} \, . \end{align} The difficulty is then to figure out the string string functions $C_{ij}^{mn}$. This has been done for all highest weight representations up to level 2 in [1] and up to level 3 in [4].
I want to thank Alex Feingold and Axel Kleinschmidt for devoting so much time to teaching me about hyperbolic Kac-Moody algebras. Also, I want to thank Benedikt König for valuable discussions.
Copyright © 2024 Hannes Malcha
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